The very basic dynamical system formulation for \(f(R)\)  gravity presented in this section has traditionally been employed in the context of inflation in the early universe [1]}, [2]}, which is mostly modelled as an exponential expansion of a matter-free universe. As we have seen, this formulation proves to be particularly well-suited for this use, as the phase space is a 2-dimensional plane with the actual physically relevant quantities as the coordinates. This advantage will disappear once we consider the global spatial curvature and/or matter (i.e., terms which has an explicit scale factor dependence). Later on, as it became clear that \(f(R)\)  gravity theories also have immense use in the late universe context as an alternative to dark energy, ignoring the global spatial curvature or the matter is no longer justifiable. This is how the other dynamical system formulations came into the picture.
